Documentation of TauSpinner approach for electroweak corrections in LHC $Z \rightarrow ll$ observables

Abstract

The LHC Standard Model Z-boson couplings measurements approach the LEP legacy precision. The calculations of electroweak (EW) corrections available for the Monte Carlo generators become of relevance. Predictions of Z-boson production and decay require classes of QED/EW/QCD corrections and separatly from the production process QCD dynamics. At the LEP time electroweak form-factors and Improved Born Approximation were introduced for non QED genuine weak and line-shape corrections. This formalism was well suited for observables, so-called doubly-deconvoluted the Z-pole region where initial- and final-state QED real and virtual emissions were treated separately or were integrated over. The approach was convenient for implementation into Monte Carlo programs for LEP, Belle-BaBar and other future $e^+e^-$ colliders and for invariant mass of outgoing lepton pair from a few GeV to well above WW and even $t{\bar{t}}$ threshold. We attempt now to profit from that, for the LHC pp and 70–150 GeV window for the outgoing lepton pair invariant mass. Our technical focus is on the EW corrections for LHC $Z \rightarrow \ell \ell$ observables. For this purpose the TauSpinner package, for the reweighting of previously generated events, is enriched with the genuine EW corrections (QED effects subtracted) of the Dizet electroweak library, taken from the LEP era KKMC Monte Carlo. Complete genuine EW $O(\alpha )$ weak loop corrections and dominant higher-order terms are taken into account. For the efficiency and numerical stability look-up tables are used. For LHC observables: the Z-boson line-shape, the outgoing leptons forward-backward asymmetry, the effective leptonic weak mixing angles and finally for the spherical harmonic expansion coefficients of the lepton distributions, corrections are evaluated. Simplified calculations of Effective Born of modified EW couplings are compared with of Improved Born Approximation of complete set of EW form-factors. Approach uses LEP precision tests definitions and thus offers consistency checks. The package can be useful to evaluate of observables precision limits and to determine which corrections are then important for LHC and FCC projects phenomenology.

1 Introduction

A theoretically sound separation of QED/EW effects between the QED emissions and genuine weak effects was essential for the phenomenology of LEP precision physics [1]. It was motivated by the structure of the amplitudes for single Z or (to a lesser degree) WW pairs production in $e^+e^-$ collisions, and by the fact that QED bremsstrahlung occurs at a different energy scale than the electroweak processes. Even more importantly, with this approach multi-loop calculations for complete electroweak sector could be avoided. The QED terms could be resumed in an exclusive exponentiation scheme implemented in Monte Carlo [2]. Note that QED corrections modify the cross-section at the peak by as much as 40%. The details of this paradigm are explained in [3]. It was obtained as a consequence of massive efforts, we will not recall them here. For the present study, the observation that spin amplitudes semi-factorize into a Born-like terms and functional factors responsible for bremsstrahlung [4] was very important.

A similar separation can be also achieved for dynamics of production process in pp collisions, which can be isolated from QED/EW corrections. It was explored recently in the case of configurations with high-$p_T$ jets associated with the Drell-Yan production of Z [5] or W bosons [6] at LHC. The potentially large electroweak Sudakov logarithmic corrections discussed in [7] (absent in our work) represent yet another class of weak effects, separable from those discussed throughout this paper. They are very small for lepton pairs with a virtuality close to the Z-boson pole mass and, if accompanied by the jet when virtuality of $\ell \ell j$ system is not much larger than 2 $M_W$. Otherwise the Sudakov corrections have to be revisited and calculation of electroweak corrections extended, even if invariant mass of the lepton pair is close to the Z mass.

To assess precisely the size and impact of genuine weak corrections to the Born-like cross section for lepton pair production with a virtuality below threshold for WW pair production, the precision calculations and programs prepared for the LEP era: KKMC Monte Carlo [8] and Dizet electroweak (EW) library, were adapted to provide pre-tabulated EW corrections to be used by LHC specific event reweighting programs like TauSpinner package [9]. Even at present KKMC Monte Carlo use Dizet version 6.21 [10, 11]. We restrict ourselves to that reference version. The TauSpinner package was initially created as a tool to correct with per-event weight longitudinal spin effects in the generated event samples including $\tau$ decays. Algorithms implemented there turned out to be of more general usage. The possibility to introduce one-loop electroweak corrections from SANC library [12] in case of Drell-Yan production of the Z-boson became available in TauSpinner since [13]. Pre-tabulation prepared for EW corrections of SANC library, was useful to introduce weights for complete spin effects at each individual event level. However no higher loop contributions were available.

TauSpinner provides a reweighting technique to modify hard process matrix elements (also matrix elements for $\tau$ decays) which were used for Monte Carlo generation. For each event no changes of any details for event kinematic configurations are introduced. The reweighting algorithm can be used for events where final state QED bremsstrahlung photons and/or high $p_T$ jets are present. For matrix element calculation used for re-weighting, some contributions such as of QED bremsstrahlung or of jet emissions have to be removed. For that purpose factorization and detailed inspection of fixed order perturbation expansion amplitudes is necessary. The most recent summary on algorithms and their applications is given in [14]. The reference explains in detail how kinematical configurations are reduced to Born-level configurations used for the correcting weights, also for electroweak corrections.¹

Used for both Tauola Univesal Interface and TauSpinner, SANC library [12] of year 2008 calculates one loop i.e. NLO electroweak corrections in two $\alpha (0)$ and $G\mu$ ($G_F$) schemes. It was found numerically insufficient for practical applications. For example, it was missing sizable $\alpha _s$ corrections to the calulated Z boson width. Two aspects of EW corrections implementation [13] had to be enhanced.

First, in [5, 6] we have studied separation of QCD higher order corrections and the Born-level spin amplitudes calculated in the adapted Mustraal lepton pair rest frame.² It is defined like for QED bremsstrahlung of Ref. [4]. The separation holds to a good approximation for the Drell-Yan processes where one or even two high $p_T$ jets are present. This frame is now used as option for EW weight calculation.

Second, the TauSpinner package and algorithms are now adapted to EW corrections from the Dizet library,³ more accurate than SANC. The EW corrections are introduced with form-factor corrections of Standard Model couplings and propagators which enter spin amplitudes of the Improved Born Approximation, used for EW weights calculation. They represent complete ${\mathcal {O}} (\alpha )$ electroweak corrections with QED contributions removed but augmented with carefully selected dominant higher order terms. This was very successful in analyses of LEP I precision physics. We attempt a similar strategy for the Z-boson pole LHC precision physics; the approach to EW corrections already attracted attention. It was used in the preliminary measurement of effective leptonic weak mixing angle recently published by ATLAS Collaboration [16].

This paper is organized as follows. In Sect. 2 we collect the main formulae of the formalism, in particular we recall the definition of the Improved Born Approximation. In Sect. 3 we present numerical results for the electroweak form-factors. Some details on commonly used EW schemes are discussed in Sect. 4, which also recall the definition of the Effective Born. In Sect. 5 we comment on the issues of using the Born approximation in pp collisions and in Sect. 6 we give more explanation why the Born approximation of the EW sector is still valid in the presence of NLO QCD matrix elements. In Sect. 7 we define the concept of EW weight which can be applied to introduce EW corrections into already existing samples, generated with Monte Carlo programs with EW LO hard process matrix elements only. In Sect. 8 we discuss, in numerical detail, EW corrections to different observables of interest for precision measurements: Z-boson line-shape, lepton forward-backward asymmetry and for coefficients of lepton spherical harmonic expansion. In this Section we include also a discussion of the effective weak mixing angle in case of pp collision. For results presented in Sect. 8 we use QCD NLO Powheg+MiNLO [17] $Z+j$ Monte Carlo sample, generated for pp collision with $\sqrt{s}$ = 8 TeV and EW LO implementation in matrix elements. Section 9 summarizes the paper.

In Appendix A details on the technical implementation of EW weight and how it can be calculated with help of the TauSpinner framework are given. In Appendix B formulae which have been implemented to allow variation of the weak mixing angle parameter of the Born spin amplitudes are discussed. In Appendix C initialization details, and options valuable for future discussions, for the Dizet library are collected.

2 Improved Born Approximation

At LEP times, to match higher order QED effects with the loop corrections of electroweak sector, the concept of electroweak form-factors was introduced [3]. This arrangement was very beneficial and enabled common treatment of one loop electroweak effects with not only higher order QED corrections including bremsstrahlung, but also to incorporate higher order loops into Z and photon propagators, see e.g. documentation of KKMC Monte Carlo [2] or Dizet [11]. Such description has its limitations for the LHC applications, but for the processes of the Drell-Yan type with a moderate virtuality of produced lepton pairs is expected to be useful, even in the case when high $p_T$ jets are present. For the LEP applications [1], the EW form-factors were used together with multi-photon bremsstrahlung amplitudes, but for the purpose of this paper we discuss their use with parton level Born processes only (no QED ISR/FSR).⁴

The terminology double-deconvoluted observable was widely used since LEP time and is explained e.g. in [18]. The so called Improved Born Approximation (IBA) [11] is employed. It absorbs some of the higher order EW corrections into a redefinition of couplings and propagators of the Born spin amplitude. This allows for straightforward calculation of doubly-deconvoluted observables like various cross-sections and asymmetries. QED effects are then removed or integrated over.

It is possible, because the excluded initial/final QCD and QED corrections form separately gauge invariant subsets of diagrams [11]. The QED subset consists of QED-vertices, $\gamma \gamma$ and $\gamma Z$ boxes and bremsstrahlung diagrams. The subset corresponding to the initial/final QCD corrections can be constructed as well. All the remaining corrections contribute to the IBA: purely EW loops, boxes and internal QCD corrections for loops (line-shape corrections). They can be split into two more gauge-invariant subsets, giving rise to two improved (or dressed) amplitudes: (i) improved $\gamma$ exchange amplitude with running QED coupling where fermion loops of low $Q^2$ contribute dominantly and (ii) improved Z-boson exchange amplitude with four complex EW form-factors: $\rho _{\ell f}$, ${{\mathscr {K}}}_{\ell }$, ${\mathscr {K}}_{f}$, ${{\mathscr {K}}}_{\ell f}$. Components of those corrections are as follows:

• Corrections to photon propagator, where fermion loops contribute dominantly the so called vacuum-polarization corrections.

• Corrections to Z-boson propagator and couplings, called EW form-factors.

• Contribution from the purely weak WW and ZZ box diagrams. They are negligible at the Z-peak (suppressed by the factor $(s-M^2_Z)/s$), but very important at higher energies. They enter as corrections to form-factors and introduce non-polynomial dependence on the $\cos$ of the scattering angle.

• Mixed $O(\alpha \alpha _s, \alpha \alpha _s^2, ...)$ corrections which originate from gluon insertions to the fermionic components of bosonic self-energies. They enter as corrections to all form-factors.

Below, to define notation we present the formula of the Born spin amplitude ${{\mathscr {A}}}^{Born}$. We recall conventions from [11]. Let us start with defining the lowest order coupling constants (without EW corrections) of the Z boson to fermions: $s^2_W = 1- M_W^2/M_Z^2=\sin \theta ^2_W$ defines weak Weinberg angle in the on-mass-shell scheme and $T_3^{\ell , f}$ third component of the isospin. The vector $v_{\ell }, v_f$ and axial $a_{\ell }, a_f$ couplings for leptons and quarks are defined with the formulae below⁵

$$\begin{aligned} v_{\ell }&= (2 \cdot T_3^{\ell } - 4 \cdot q_{\ell } \cdot s^2_W)/\Delta , \nonumber \\ v_f&= (2 \cdot T_3^f - 4 \cdot q_f \cdot s^2_W)/\Delta , \nonumber \\ a_{\ell }&= (2 \cdot T_3^{\ell } )/\Delta , \nonumber \\ a_f&= (2 \cdot T_3^f )/\Delta . \end{aligned}$$

(1)

where

$$\begin{aligned} \Delta = \sqrt{ 16 \cdot s^2_W \cdot (1 - s^2_W)} , \end{aligned}$$

(2)

and $q_f$, $q_l$ denote charge of incoming fermion (quark) and outgoing lepton. With this notation, the ${{\mathscr {A}}}^{Born}$ spin amplitude for the $q {\bar{q}} \rightarrow Z/\gamma ^* \rightarrow \ell ^+ \ell ^-$ can be written as:

$$\begin{aligned}&{{\mathscr {A}}}^{Born}\nonumber \\&\quad = \frac{\alpha }{s}\ \ \{ [{\bar{u}} \gamma ^{\mu } v g_{\mu \nu } {\bar{v}} \gamma ^{\nu } u] \cdot ( q_{\ell } \cdot q_f) \cdot \chi _{\gamma }(s)\nonumber \\&\qquad + [{\bar{u}} \gamma ^{\mu } v g_{\mu \nu } {\bar{\nu }} \gamma ^{\nu } u \cdot ( v_{\ell } \cdot v_f ) \nonumber \\&\qquad + {\bar{u}} \gamma ^{\mu } v g_{\mu \nu } {\bar{\nu }} \gamma ^{\nu } \gamma ^5 u \cdot (v_{\ell } \cdot a_f)\nonumber \\&\qquad + {\bar{u}} \gamma ^{\mu } \gamma ^5 v g_{\mu \nu } {\bar{\nu }} \gamma ^{\nu } u \cdot (a_{\ell } \cdot v_f)\nonumber \\&\qquad + {\bar{u}} \gamma ^{\mu } \gamma ^5 v g_{\mu \nu }{\bar{\nu }} \gamma ^{\nu } \gamma ^5 u \cdot (a_{\ell } \cdot a_f) ] \cdot \chi _Z (s)\} , \end{aligned}$$

(3)

where u, v denote fermion spinors and, $\alpha$ stands for QED coupling constant. The Z-boson and photon propagators are defined respectively as:

$$\begin{aligned} \chi _{\gamma }(s)= & {} 1,\end{aligned}$$

(4)

$$\begin{aligned} \chi _Z(s)= & {} \frac{G_{\mu } \cdot M_{z}^2 \cdot \Delta ^2 }{\sqrt{2} \cdot 8 \pi \cdot \alpha }\cdot \frac{s}{s - M_Z^2 + i \cdot \Gamma _Z \cdot s/M_Z}. \end{aligned}$$

(5)

For the IBA, we redefine vector and axial couplings and introduce EW form-factors $\rho _{\ell f}(s,t), {\mathscr {K}}_{\ell }(s,t)$, ${{\mathscr {K}}}_f(s,t)$, ${{\mathscr {K}}}_{\ell f} (s,t)$ as follows:

$$\begin{aligned} v_{\ell }= & {} (2 \cdot T_3^{\ell } - 4 \cdot q_{\ell } \cdot s^2_W \cdot {{\mathscr {K}}}_{\ell }(s,t))/\Delta , \nonumber \\ v_f= & {} (2 \cdot T_3^f - 4 \cdot q_f \cdot s^2_W \cdot {{\mathscr {K}}}_f(s,t))/\Delta , \nonumber \\ a_{\ell }= & {} (2 \cdot T_3^{\ell } )/\Delta , \nonumber \\ a_f= & {} (2 \cdot T_3^f )/\Delta . \end{aligned}$$

(6)

Normalization correction $Z_{V_{\Pi }}$ to the Z-boson propagator is defined as

$$\begin{aligned} Z_{V_{\Pi }} = \rho _{\ell f}(s,t) \ . \end{aligned}$$

(7)

Re-summed vacuum polarization corrections $\Gamma _{V_{\Pi }}$ to the $\gamma ^*$ propagator are expressed as

$$\begin{aligned} \Gamma _{V_{\Pi }} = \frac{1}{ 2 - (1 + \Pi _{\gamma \gamma }(s))}, \end{aligned}$$

(8)

where $\Pi _{\gamma \gamma }(s)$ denotes vacuum polarization loop corrections of virtual photon exchange. Both $\Gamma _{V_{\Pi }}$ and $Z_{V_{\Pi }}$ are multiplicative correction factors. The $\rho _{\ell f}(s,t)$ could be also absorbed as multiplicative factor into the definition of vector and axial couplings.

The EW form-factors $\rho _{\ell f}(s,t), {\mathscr {K}}_{\ell }(s,t)$, ${{\mathscr {K}}}_f(s,t)$, ${{\mathscr {K}}}_{\ell f}(s,t)$ depend on two Mandelstam invariants (s, t) due to contributions of the WW and ZZ boxes. The Mandelstam variables satisfy the identity

$$\begin{aligned} s+t+u = 0 \ \ \ where \ \ \ \ t = -\frac{s}{2}(1 - \cos \theta ) \end{aligned}$$

(9)

and $\cos \theta$ is the cosine of the scattering angle, i.e. the angle between incoming and outgoing fermion directions.

Note, that in this approach the mixed EW and QCD loop corrections, originating from gluon insertions to fermionic components of bosonic self-energies, are included in $\Gamma _{V_{\Pi }}$ and $Z_{V_{\Pi }}$.

One has to pay special attention to the angle dependent product of the vector couplings. The corrections break factorization, formula (3), of the couplings into ones associated with either Z boson production or decay. The mixed term has to be added:

$$\begin{aligned} vv_{\ell f} =&\frac{1}{v_{\ell } \cdot v_f} \Big [ ( 2 \cdot T_3^{\ell }) (2 \cdot T_3^f) - 4 \cdot q_{\ell } \cdot s^2_W \cdot {\mathscr {K}}_f(s,t)( 2 \cdot T_3^{\ell })\nonumber \\&- 4 \cdot q_f \cdot s^2_W \cdot {{\mathscr {K}}}_{\ell }(s,t) (2 \cdot T_3^f) \nonumber \\&+ (4 \cdot q_{\ell } \cdot s^2_W) (4 \cdot q_f \cdot s^2_W) {{\mathscr {K}}}_{\ell f}(s,t)\Big ] \frac{1}{\Delta ^2}. \end{aligned}$$

(10)

Finally, we can write the spin amplitude for Born with EW corrections, ${{\mathscr {A}}}^{Born+EW}$, as:

$$\begin{aligned} { {\mathscr {A}}}^{Born+EW}= & {} \frac{\alpha }{s} \{ [{\bar{u}} \gamma ^{\mu } v g_{\mu \nu } {\bar{v}} \gamma ^{\nu } u] \cdot ( q_{\ell } \cdot q_f)] \cdot \Gamma _{V_{\Pi }} \cdot \chi _{\gamma }(s) \nonumber \\&+ [{\bar{u}} \gamma ^{\mu } v g_{\mu \nu } {\bar{\nu }} \gamma ^{\nu } u \cdot ( v_{\ell } \cdot v_f \cdot vv_{\ell f})\nonumber \\&+ {\bar{u}} \gamma ^{\mu } v g_{\mu \nu } {\bar{\nu }} \gamma ^{\nu } \gamma ^5 u \cdot (v_{\ell } \cdot a_f) \nonumber \\&+ {\bar{u}} \gamma ^{\mu } \gamma ^5 v g_{\mu \nu } {\bar{\nu }} \gamma ^{\nu } u \cdot (a_{\ell } \cdot v_f) \nonumber \\&+ {\bar{u}} \gamma ^{\mu } \gamma ^5 v g_{\mu \nu }{\bar{\nu }} \gamma ^{\nu } \gamma ^5 u \cdot (a_{\ell } \cdot a_f) ] \cdot Z_{V_{\Pi }}\cdot \chi _Z (s)\}.\nonumber \\ \end{aligned}$$

(11)

The EW form-factor corrections: $\rho _{\ell f}, {{\mathscr {K}}}_{\ell }, {{\mathscr {K}}}_f, {{\mathscr {K}}}_{\ell f}$ can be calculated using the Dizet library. This library invokes also calculation of vacuum polarization corrections to the photon propagator $\Pi _{\gamma \gamma }$. For the case of pp collisions we do not introduce QCD corrections to vector and axial couplings of incoming fermions. They are assumed to be included elsewhere as a part of the QCD NLO calculations for the initial parton state, including convolution with proton structure functions.

The Improved Born Approximation uses the spin amplitude ${{\mathscr {A}}}^{Born+EW}$ of Eq. (11) and $2 \rightarrow 2$ body kinematics to define the differential cross-section with EW corrections for $q {\bar{q}} \rightarrow Z/\gamma ^* \rightarrow l l$ process. The formulae presented above very closely follow the approach taken for implementation⁶ of EW corrections to KKMC Monte Carlo [2].

3 Electroweak form-factors

For the calculation of EW corrections, we use the Dizet library, as of the 2010 KKMC Monte Carlo [2] version. For this and related projects, massive theoretical effort was necessary. Simultaneous study of several processes, like of $\mu ^+\mu ^-$, $u {\bar{u}}$, $d {\bar{d}}$, $\nu {\bar{\nu }}$ production in $e^+e^-$ collisions and also in $p {\bar{p}}$ initiated parton processes, like at Tevatron, was performed. Groups of diagrams for the $Z/\gamma ^*$ propagators, production and decay vertices could be identified and incorporated into form-factors. The core of the Dizet library relies on such separation. It also opened the possibility that for one group of diagrams, such as vacuum polarizations, higher order contributions could be included while for others were not. That was particularly important for quark contributions to vacuum polarizations. Otherwise, the required precision would not be achieved. The above short explanation only indicates fundamental importance of the topic, we delegate the reader to Refs. [2, 19, 20] and experimental papers of LEP and Tevatron experiments quoting these papers.

The interface in KKMC prepares look-up tables with EW form-factors and vacuum polarization corrections. The tabulation grid granularity and ranges of the centre-of-mass energy of outgoing leptons and lepton scattering angle are adapted to variation of the tabulated functions. Theoretical uncertainties on the predictions for EW form-factors have been estimated in times of LEP precision measurements, in the context of either benchmark results like [18] or specific analyses [3]. The predictions are now updated with the known Higgs boson and top-quark masses. In the existing code of the Dizet library, certain types of the corrections or options of the calculations of different corrections can be switched off/on. In Appendix C, we show in Table 11 an almost complete list of options useful for discussions. We do not attempt to estimate the size of theoretical uncertainties, delegating it to the follow up work in the context of LHC EW Precision WG studies. The other versions of electroweak calculations, like of [12, 21], can and should be studied then as well. Already now the precision requirements of LHC experiments [16] are comparable to those of individual LEP measurements, but phenomenology aspects are more involved.

3.1 Input parameters to Dizet

The Dizet package relies on the so called on-mass-shell (OMS) normalization scheme [19, 20] but modifications are present. The OMS uses the masses of all fundamental particles, both fermions and bosons, the electromagnetic coupling constant $\alpha (0)$ and the strong coupling $\alpha _s(M_Z^2)$. The dependence on the ill-defined masses of the light quarks u, d,c, s and b is solved by dispersion relations, for details see [11]. Another exception is the W-boson mass $M_W$, which still can be predicted with better theoretical accuracy than experimentally measured. The Fermi constant $G_{\mu }$ is precisely known from $\mu$-decay. For this reason, $M_W$ was usually, in time of LEP analyses, replaced by $G_{\mu }$ as an input.

The knowledge about the hadronic vacuum polarization is contained in $\Delta \alpha _h^{(5)}(s)$, which is used as external, easy to change, parametrization. It can be either computed from quark masses or, preferably, fitted to experimental low energy $e^+ e^- \rightarrow hadrons$ data.

The $M_W$ is calculated iteratively from the equation

$$\begin{aligned} M_W = \frac{M_Z}{\sqrt{2}} \sqrt{ 1 + \sqrt{ 1 - \frac{ 4 A^2_0}{M^2_Z ( 1 - \Delta r)}}}, \end{aligned}$$

(12)

where

$$\begin{aligned} A_0 = \sqrt{ \frac{\pi \alpha (0)}{\sqrt{2}G_{\mu }}}. \end{aligned}$$

(13)

The Sirlin’s parameter $\Delta r$ [22]

$$\begin{aligned} \Delta r = \Delta \alpha (M_Z^2) + \Delta r_{EW} \end{aligned}$$

(14)

is also calculated iteratively, and the definition of $\Delta r_{EW}$ involves re-summation and higher order corrections. This term implicitly depends on $M_W$ and $M_Z$, and the iterative procedure is needed. The re-summation term in formula (14) is not formally justified by renormalisation group arguments, the correct generalization is to compute higher order corrections, see discussion in [11].

Note that once the $M_W$ is recalculated from formula (12), the lowest order Standard Model relationship between the weak and electromagnetic couplings

$$\begin{aligned} G_{\mu } = \frac{\pi \alpha }{\sqrt{2} M^2_W \sin ^2\theta _W} \end{aligned}$$

(15)

is not fulfilled anymore, unless the $G_{\mu }$ is redefined away from the measured value. This is an approach of some EW LO schemes, but not the one used by Dizet. It requires therefore the complete expression for $\chi _Z(s)$ propagator in spin amplitude of Eq.  (11), as defined by formula (5).

In the OMS renormalisation scheme the weak mixing angle is defined uniquely through the gauge-boson masses:

$$\begin{aligned} \sin ^2\theta _W = s^2_W = 1 - \frac{M^2_W}{M^2_Z}. \end{aligned}$$

(16)

With this scheme, measuring $\sin ^2\theta _W$ will be equivalent to indirect measurement of $M^2_W$ through the relation (16).

In Table 1 we collect numerical values for all parameters used in the presented below evaluations. Note that formally they are not representing EW LO scheme, as the relation (15) is not obeyed. The $M_W$ in (16) is recalculated with (12) but $G_{\mu }$, $M_Z$ remain unchanged.

Table 1 The Dizet initialization: masses and couplings. The calculated $M_W$ and $s^2_W$ are shown also

3.2 The EW form-factors

Real parts of the $\rho _{\ell f}(s,t)$, ${{\mathscr {K}}}_f(s,t)$, ${{\mathscr {K}}}_{\ell }(s,t)$, ${{\mathscr {K}}}_{\ell f}(s,t)$ EW form-factors are shown in Fig. 1 for a few values of $\cos \theta$, the angle between directions of the incoming quark and the outgoing lepton, calculated in the outgoing lepton pair centre-of-mass frame. Eq. (9) relates Mandelstam variables (s, t) to the invariant mass and $\cos \theta$. The $\cos \theta$ dependence of the box correction is more sizable for the up-quarks.

Note, that at the Z-boson peak, Born-like couplings are only weakly modified; form-factors are close to 1 and of no numerically significant angular dependence. At lower virtualities corrections are relatively larger because the Z-boson contributions are non resonant and thus smaller. In this phase-space region the Z-boson is itself dominated by the virtual photon contribution. Above the peak, the WW and later also ZZ boxes contributions become sizable, the dependence on the $\cos \theta$ appears; contributions become gradually doubly resonant and sizable.

Fig. 1

Real parts of the $\rho _{e, up}$, ${{\mathscr {K}}}_{e}$, ${\mathscr {K}}_{up}$ and ${{\mathscr {K}}}_{e, up}$ EW form-factors of $q {\bar{q}} \rightarrow Z \rightarrow ee$ process, as a function of $\sqrt{s}$ and for the few values of $\cos \theta$. For the up-type quark flavour, left side plots are collected and for the down-type the right side plots. Note, that ${{\mathscr {K}}_e}$ depends on the flavour of incoming quarks

3.3 Running $\alpha (s)$

Fermionic loop insertion of the photon propagator, i.e. vacuum polarization corrections, are summed together as a multiplicative factor $\Gamma _{V_{\Pi }}$, Eq. (8), for the photon exchange in Eq. (11). But it can be interpreted as the running QED coupling:

$$\begin{aligned} \alpha (s) = \frac{\alpha (0)}{1 - \Delta \alpha _h^{(5)}(s) - \Delta \alpha _{\ell }(s) - \Delta \alpha _t(s) - \Delta \alpha ^{\alpha \alpha _s}(s)}.\nonumber \\ \end{aligned}$$

(17)

The hadronic contribution at $M_Z$ is a significant [11] correction: $\Delta \alpha _h^{(5)}(M_Z^2) = 0.0280398$. It is calculated in the five flavour scheme with use of dispersion relation and input from low energy experiments. We will continue to use LEP times parametrization, while the most recent measured $\Delta \alpha _h^{(5)}(M_Z^2) = 0.02753 \pm 0.00009$ [23]. The changed value modifies predicted form-factors, in particular the effective leptonic mixing angle $\sin ^2\theta _{eff}^{lep}(M_Z^2)=Re{({{\mathscr {K}}}_l(M_Z^2))} s^2_W$ is shifted by almost $20 \cdot 10^{-5}$ closer to the measured LEP value. This is not included in the numerical results presented as we consistently remain with the defaults used in KKMC.

The leptonic loop contribution $\Delta \alpha _{\ell }(s)$ is calculated analytically up to the 3-loops, and is a comparably significant correction, $\Delta \alpha _{\ell }(M_Z^2)$ = 0.0314976. The other contributions are very small.

Figure 2 shows the vacuum polarization corrections to the $\chi _{\gamma }(s)$ propagator, directly representing the ratio $\alpha (s)/\alpha (0)$ of Eq. (17).

4 EW input schemes and effective born

Formally, at the lowest EW order, only three independent parameters can be set, other are calculated following the structure of $SU(2) \times U(1)$ group from Standard Model constraints. Formula (15) represents one of such constraints. Following report [24], the most common choices at hadron colliders are: $G_{\mu }$ scheme $(G_{\mu }, M_Z, M_W)$ and $\alpha (0)$ scheme $(\alpha (0)$, $M_Z, M_W)$. There exists by now a family of different modifications of the $G_{\mu }$ scheme, see discussion in [24], and they are considered as preferred schemes for hadron collider physics.⁷

Let us recall, that the calculations of EW corrections available in Dizet work with a variant of the $\alpha (0)$ scheme. It is defined by the input parameters $(\alpha (0), G_{\mu }, M_Z)$. Then $M_W$ is calculated iteratively from formula (12) and $s^2_W$ of Eq. (16) uses that value of $M_W$. This formally brings it beyond EW LO scheme. The numerical value of $s_W^2$ calculated from (16) does not fulfill the EW LO relation (15) anymore.

Fig. 2

The vacuum polarization ($\alpha (s)/\alpha (0)$) correction of $\gamma$ propagator, Eq. (17)

At this point we introduce two options for the Effective Born spin amplitudes parametrization, which works well for parametrizing EW corrections near the Z-pole and denote them respectively as LEP and LEP with improved norm.:

• The LEP parametrization uses formula (11) for spin amplitude but with $\alpha (s) = \alpha (M_Z^2) = 1./128.8667$, $s^2_W = \sin ^2\theta _W^{eff} (M_Z^2) = 0.23152$, i.e. as measured at the Z-pole and reported in [25]. All form-factors are set to 1.0.

• The LEP with improved norm. parametrization also uses formula (11) for spin amplitude with parameters set as for LEP parametrization. All form-factors are set to 1, but $\rho _{\ell f} = 1.005$. This corresponds to the measured $\rho (M_Z^2)$ = 1.005, as reported in [25].

Table 2 collects initialization constants of EW schemes relevant for our discussion. We specify parameters which enter formula (11) for Born spin amplitudes used for: (i) actual MC events generation,⁸ (ii) the EW LO $\alpha (0)$ scheme, (iii) effective Born (LEP) parametrization and (iv) effective Born (LEP with improved norm.). In each case parameters are chosen such that the SM relation, formula (18), is obeyed.

In the Improved Born Approximation complete $O(\alpha )$ EW corrections, supplemented by selected higher order terms, are handled thanks to s-, t-dependent form-factors, which multiply couplings and propagators of the usual Born expressions. Instead, the Effective Born absorbs the bulk of EW corrections into a redefinition of a few fixed parameters (i.e. couplings).

Table 2 The EW parameters used for: (i) MC events generation, (ii) the EW LO $\alpha (0)$ scheme, (iii) effective Born spin amplitude around the Z-pole and (iv) effective Born with improved normalization. In each case parameters are chosen such that the SM relation, formula (18), is obeyed. The $G_{\mu }$ = $1.166389 \cdot 10^{-5}$ $\hbox {GeV}^{-2}$, $M_Z$ = 91.1876 GeV and ${{\mathscr {K}}}_f, {{\mathscr {K}}}_e, {{\mathscr {K}}}_{\ell f}$ = 1 are taken

In the following, we will systematically compare predictions obtained with the EW corrections and those calculated with LEP or LEP with improved norm. approximations. As we will see, effective Born with LEP with improved norm. works very well around Z-pole both for the line-shape and forward-backward asymmetry.

5 Born kinematic approximation and pp scattering

The solution to define Born-like parton level kinematics for pp scattering process is encoded in the TauSpinner package [14]. It does not exploit hard-process, so-called history entries which only sometimes are stored for the generated events. In particular, the flavour and momenta of the incoming partons have to be emulated from the kinematics of final states and incoming protons momenta. Probabilities calculated from parton level cross-sections and PDFs weight all possible contributions. Let us now recall briefly principles and choices for optimization.

5.1 Average over incoming partons flavour

The parton level Born cross-section $\sigma ^{q {\bar{q}}}_{Born}({\hat{s}}, \cos \theta )$ has to be convoluted with the structure functions, and summed over all possible flavours of incoming partons and all possible helicity states of outgoing leptons. The lowest order formula⁹ is given below

$$\begin{aligned}&d\sigma _{Born}( x_1, x_2, {\hat{s}}, \cos \theta ) \nonumber \\&\quad =\sum _{q_f, {\bar{q}}_f} [ f^{q_f}(x_1,...)f^{{\bar{q}}_f}(x_2,...) d\sigma ^{q_f {\bar{q}}_f}_{Born}( {\hat{s}}, \cos \theta ) \nonumber \\&\qquad + \ f^{{\bar{q}}_f}(x_1,...)f^{ q_f}(x_2,...) d\sigma ^{ q_f \bar{q}_f}_{Born}( {\hat{s}}, -\cos \theta ) ] , \end{aligned}$$

(19)

where $x_1$, $x_2$ denote fractions of incoming protons momenta carried by the corresponding parton, ${\hat{s}} = x_1\ x_2\ s$ and $f/{\bar{f}}$ denotes parton (quark-/anti-quark) density functions. We assume that kinematics is reconstructed from four-momenta of the outgoing leptons. The incoming quark and anti-quark may come respectively either from the first and second proton or reversely from the second and first. Both possibilities are taken into account¹⁰ by the two terms of (19). The sign in front of $\cos \theta$, the cosine of the scattering angle, is negative for the second term. Then the parton of the first incoming proton which carries $x_1$ and follows the direction of the z-axis is an anti-quark, not a quark. The formula is used for calculating the differential cross-section $d\sigma _{Born}( x_1, x_2, {\hat{s}}, \cos \theta )$ of each analyzed event, regardless if its kinematics and flavours of incoming partons may be available from the event history entries or not. The formula can be used to a good approximation in case of NLO QCD spin amplitudes. The momenta of outgoing leptons are used to construct effective kinematics of the Drell-Yan production process and decay, without the need of information on parton-level hard-process itself. Born-like kinematics can be constructed, as we will see later, even for events of quark-gluon or gluon-gluon parton level collisions (as inspected for test in the event history entries) too.

5.2 Effective beams kinematics

The $x_1, x_2$ are calculated from the kinematics of outgoing leptons, following formulae of [15]

$$\begin{aligned} x_{1,2} = \frac{1}{2}\ {\Big (}\ \pm \frac{p_z^{ll}}{ E} + \sqrt{ (\frac{p_z^{ll}}{ E})^2 + \frac{m^2_{ll}}{ E^2}} \; \; {\Big )} , \end{aligned}$$

(20)

where E denotes energy of the proton beam and $p_z^{\ell \ell }$ denotes z-axis momentum of outgoing lepton pair in the laboratory frame and $m_{ll}$ lepton pair virtuality. Note that this formula can be used, as approximation, for the events with hard jets too.

5.3 Definition of the polar angle

For the polar angle $\cos \theta$, of factorized Born level $q \bar{q} \rightarrow Z \rightarrow \ell \ell$ process, weighted average of the outgoing leptons angles with respect to the beams’ directions, denoted as $\cos \theta ^*$, was used. In [28] it was found helpful to compensate the effect of initial state hard bremsstrahlung photons of $e^+e^- \rightarrow Z n\gamma$, $Z \rightarrow \ell \ell m\gamma$, where $m,\; n$ denote the number of accompanying photons. Extension to pp collisions required to take both options in Eq. (19) into account; when the z-axis is parallel- and anti-parallel to the incoming quark.

For the further calculation, boost of all four-momenta (also of incoming beams) into the rest frame of the lepton pair need to be performed. The $\cos \theta ^{*}$ is then calculated from

$$\begin{aligned} \cos \theta _1 = \frac{\tau _x^{(1)} b_x^{(1)} + \tau _y^{(1)} b_y^{(1)} + \tau _z^{(1)} b_z^{(1)}}{ | \mathbf {\tau }^{(1)}| |{{\varvec{b}}}^{(1)}|},\nonumber \\ \cos \theta _2 = \frac{\tau _x^{(2)} b_x^{(2)} + \tau _y^{(2)} b_y^{(2)} + \tau _z^{(2)} b_z^{(2)}}{ | \mathbf {\tau }^{(2)}| |{{\varvec{b}}}^{(2)}|}, \end{aligned}$$

(21)

as follows:

$$\begin{aligned} \cos \theta ^* = \frac{\cos \theta _1 \sin \theta _2 + \cos \theta _2 \sin \theta _1}{\sin \theta _1 + \sin \theta _2} \end{aligned}$$

(22)

where $\mathbf {\tau }^{(1)}, \mathbf {\tau }^{(2)}$ denote 3-vectors of outgoing leptons and ${{\varvec{b}}}^{(1)}, {{\varvec{b}}}^{(2)}$ denote 3-vectors of incoming beams’ four-momenta.

The polar angle definition, Eq. (22), is at present the TauSpinner default. For tests we have used variants; Mustraal [4] and Collins-Soper [29] frames, which differ when high $p_T$ jets are present. We will return later to the frame choice, best suitable when NLO QCD corrections are included in the production process of generated events.

6 QCD corrections and angular coefficients

For the Drell-Yan production [30] one can separate QCD and EW components of the fully differential cross-section and describe the $Z/\gamma ^* \rightarrow \ell \ell$ sub-process with lepton angular ($\theta , \phi$) dependence

$$\begin{aligned} \frac{ d\sigma }{dp_T^2 dY d\Omega } = \Sigma _{ \alpha =1}^{9} g_{ \alpha }( \theta , \phi ) \frac{3}{ 16 \pi } \frac{d \sigma ^{\alpha }}{ dp_T^2 dY}, \end{aligned}$$

(23)

where the $g_{ \alpha }( \theta , \phi )$ denotes second order spherical harmonics, multiplied by normalization constants and $d \sigma ^{\alpha }$ denotes helicity cross-sections, for each of nine helicity configurations of $q {\bar{q}} \rightarrow Z/\gamma ^* \rightarrow \ell \ell$. The polar and azimuthal ($\theta$ and $\phi$) angles of $d\Omega = d \cos \theta d\phi$ are defined in the Z-boson rest-frame. The $p_T$, Y denote laboratory frame transverse momenta and rapidity of the intermediate $Z/\gamma ^*$-boson. Thanks to the effort [31,32,33] from the early 90’s one expects such factorization to break with non-logarithmic ${{\mathcal {O}}}(\alpha _s^2) \sim 0.01$ QCD corrections¹¹ only.

There is some flexibility for the Z-boson rest frame z-axis choice. The most common, so called helicity frame, is to take the Z-boson laboratory frame momentum. For the Collins-Soper frame it is defined from directions of the two beams in the Z-boson rest frame and is signed with the Z-boson $p_z$ laboratory frame sign.

Equation (23) with explicit spherical harmonics and coefficients reads

$$\begin{aligned}&\frac{ d\sigma }{dp_T^2 dY d \cos \theta d\phi } = \frac{3}{ 16 \pi } \frac{d \sigma ^{ U+ L }}{ dp_T^2 dY} [ (1 + \cos ^2\theta ) \nonumber \\&\quad + 1/2\ A_0 (1 - 3 \cos ^2\theta ) + A_1 \sin {2\theta }\cos \phi \nonumber \\&\quad + 1/2\ A_2 \sin ^2\theta \cos ( 2 \phi ) + A_3 \sin \theta \cos \phi + A_4 \cos \theta \nonumber \\&\quad + A_5 \sin ^2 \theta \sin ( 2 \phi ) + A_6 \sin {2\theta } \sin \phi + A_7 \sin \theta \ \sin \phi ],\nonumber \\ \end{aligned}$$

(24)

where $d \sigma ^{U+ L}$ denotes the unpolarised differential cross-section (notation used in several papers of the 80’s). The coefficients $A_i(p_T, Y)$ are related to ratios of definite intermediate state helicity contributions to the $d \sigma ^{ U+ L }$ cross-sections. The first term of the polynomial expansion is $(1 + \cos ^2\theta )$ because intermediate boson is of the spin 1.

The dynamics of the production process is hidden in the angular coefficients $A_i (p_T, Y)$. In particular, all the hadronic physics is described implicitly by the angular coefficients and it decouples from the well understood leptonic and intermediate boson physics.

For the present paper, of particular interest are coupling constants present in coefficients $A_i$ of Eq. (24) representing ratios of the so-called helicity cross sections [31,32,33]:

$$\begin{aligned} \sigma ^{U+L }\sim & {} (v_{\ell }^2 + a_{\ell }^2)(v_{q}^2 + a_{q}^2), \nonumber \\ A_0, A_1, A_2\sim & {} 1 , A_3, A_4 \sim \frac{ v_{\ell } a_{\ell } v_q a_q}{(v_{\ell }^2 + a_{\ell }^2)(v_{q}^2 + a_{q}^2)} , \nonumber \\ A_5, A_6\sim & {} \frac{(v_{\ell }^2 + a_{\ell }^2) ( v_q a_q)}{(v_{\ell }^2 + a_{\ell }^2)(v_{q}^2 + a_{q}^2)}, \nonumber \\ A_7\sim & {} \frac{ v_{\ell } a_{\ell } ( v_q^2 + a_q^2)}{(v_{\ell }^2 + a_{\ell }^2)(v_{q}^2 + a_{q}^2)}. \end{aligned}$$

(25)

Integration¹² over the azimuthal angle $\phi$ reduces Eq. (24) to

$$\begin{aligned}&\frac{ d\sigma }{dp_T^2 dY d \cos \theta } = \frac{3}{ 8 \pi } \frac{d \sigma ^{ U+ L }}{ dp_T^2 dY} [ (1 + \cos ^2\theta )\nonumber \\&\quad + 1/2\ A_0 (1 - 3 \cos ^2\theta ) + A_4 \cos \theta ]. \end{aligned}$$

(26)

Both Eqs. (24) and (26) are valid in any rest frame of the outgoing lepton pairs, however the $A_i(p_T, Y)$ are frame dependent. The Collins-Soper frame is the most convenient and usual choice for the analyses dedicated to QCD dynamics. In this frame, in the low $p_T$ limit, $A_4$ is the only non-zero coefficient. It carries direct information on the EW couplings, as can be concluded from formulae (25). All other coefficients depart from zero with increasing $p_T$ while at the same time $A_4$ gradually decreases.

Due to different transfer dependence of the Z and $\gamma ^*$ propagators, the $A_i$ vary with $m_{ll}$. The $A_i$ dependence on $(p_T,Y)$, expressing production dynamics, differ with the frame definition variants of distinct coordinate system orientations. For the studies of EW couplings, it is convenient when the lepton-pair rest-frame definition absorbs effects of production dynamics partly into the z-axis choice. Then, those $A_i$ coefficients which are proportional to the product of EW vector and axial couplings remain non-zero over the full range of $p_T$. Promising for that purpose frame was developed at LEP times for the Mustraal Monte Carlo program [4]. Recently, an extension of this Mustraal frame, for the case of hadron-hadron collisions, was introduced and discussed in [5]. As shown in that paper, both Collins-Soper and Mustraal frames are equivalent in the $p_T = 0$ limit. Then $A_4$ is the only non-zero coefficient for both frames and is also numerically very close. With increasing $p_T$, in the Mustraal frame $A_4$ remains as the only sizably non-zero coefficient, while several $A_i$ coefficients depart from zero with the Collins-Soper frame.

In the collision of the same-charge protons the careful choice for the z-axis orientation is necessary for the $A_4$ coefficient to remain non-zero. For the Collins-Soper frame, the z-axis follows the direction of the intermediate Z-boson in the laboratory frame. In case of the Mustraal frame the choice of the sign is made stochastically using information of the system of leptons and outgoing accompanying visible jets. For details see [5], alternatively the same sign choice for the z-axis as in the Collins-Soper case, can be used.

The shape of $A_i$ coefficients as a function of laboratory frame Z-boson transverse momenta $p_T$ depends on the choice of lepton pairs rest-frame. In Fig. 3, $A_i$ coefficients of the Collins-Soper and Mustraal frames are shown. As intended, even for large $p_T$, with this frame, only $A_4$ coefficient is sizably non-zero.

Fig. 3

The $A_i$ coefficients for $Z\rightarrow e^+e^-$ in lepton pair invariant mass range $80< m_{ee} < 100$ GeV. The $Z+j$ production process in pp collisions at 8 TeV centre-of-mass energy, was used for the sample generation with Powheg+MiNLO Monte Carlo. The $A_i$ coefficients are calculated in the Collins-Soper and Mustraal frames with moments method [32]

7 Concept of the EW weight

The EW corrections enter the $\sigma _{Born}( {\hat{s}}, \cos \theta )$ through the definition of the vector and axial couplings, also photon and Z-boson propagators. They modify normalization of the cross-sections, the line-shape of the Z-boson peak, polarization of the outgoing leptons and asymmetries.

Given that, we were able to factorize QCD and EW components of the cross-section to a good approximation and define per-event weights which specifically correct for EW effects. Such a weight may modify events generated with EW LO to the ones including the EW corrections. This is very much the same idea as already implemented in TauSpinner for introducing corrections for other effects: spin correlations, production process, etc.

The per-event $wt^{EW}$ is defined as ratio of the Born-level cross-sections with and without EW corrections

$$\begin{aligned} wt^{EW} = \frac{ d\sigma _{Born+EW}( s, \cos \theta )}{d\sigma _{Born}( s, \cos \theta )}, \end{aligned}$$

(27)

where $\cos \theta$ can be taken according to $\cos \theta ^*$, $\cos \theta ^{Mustraal}$ (Mustraal frame) or $\cos \theta ^{CS}$ (Collins-Soper frame) prescription. For most events, the three choices will lead to numerically very close values for $\cos \theta$ and thus resulting $wt^{EW}$. The difference originates from distinct $\cos \theta$ dependence of Z and $\gamma ^*$ exchange amplitudes and not only from electroweak boxes. The $wt^{EW}$ allows for flexible implementation of the EW corrections using TauSpinner framework and form-factors calculated e.g. with Dizet.

The formula for $wt^{EW}$ can be used to re-weight from one EW LO scheme to another too. In that case, both the numerator and denominator of Eq. (27) will use lowest order $d\sigma _{Born}$, calculated in different EW schemes¹³ though.

8 EW corrections to doubly-deconvoluted observables

Now that all components needed for calculation of $wt^{EW}$ are explained, we can present results for selected examples of doubly-deconvoluted observables around the Z-pole.

The Powheg+MiNLO Monte Carlo, with NLO QCD and LO EW matrix elements, was used to generate $Z+j$ events with $Z \rightarrow e^+ e^-$ decays in pp collisions at 8 TeV. No selection was applied to generated events, except for an outgoing electron pair invariant mass range of $70< m_{ee} < 150$ GeV. For events generation, the EW parameters as shown in left-most column of Table 2 were used. It is often used as a default for phenomenological studies at LHC. The $\alpha$ and $s^2_W$ close to the ones of $\overline{\mathrm MS}$ scheme discussed in [25] were taken. Note that they do not coincide accurately with the precise LEP experiments measurements at the Z-pole [1].

To quantify the effect of the EW corrections, we re-weight events generated, to EW LO with the scheme used by the Dizet: Table 2 second column. Only then we gradually introduce EW corrections and form-factors calculated with that library. For each step, the appropriate numerator of the $wt^{EW}$ is calculated, while for the denominator the EW LO ${{\mathscr {A}}}^{Born}$ matrix element Eq. (3) is used; parameters as in the left-most column of Table 2. The sequential steps, in which we illustrate effects of EW corrections are given below:

• 1. Re-weight with $wt^{EW}$, from EW LO scheme used for MC events generation to EW LO scheme with $s^2_W$= 0.21215, Table 2 second column. The ${{\mathscr {A}}}^{Born}$ matrix element, Eq. (3), is used¹⁴ for calculating numerator of $wt^{EW}$.

• 2. As in step (1), but include EW corrections to $M_W$, effectively changing to $s^2_W$= 0.22352 in calculation of $wt^{EW}$. Relation, formula (15), is not obeyed anymore.

• 3. As in step (2), but include EW loop corrections to the normalization of Z-boson and $\gamma ^*$ propagators, i.e. QCD/EW corrections to $\alpha (0)$ and $\rho _{\ell f}(s)$ form-factor calculated without box corrections. The ${{\mathscr {A}}}^{Born+EW}$, Eq. (11), is used for calculating numerator of $wt^{EW}$.

• 4. As in step (3), but include EW corrections to Z-boson vector couplings: ${{\mathscr {K}}}_f, {{\mathscr {K}}}_l, {{\mathscr {K}}}_{\ell f}$, calculated without box corrections. The ${{\mathscr {A}}}^{Born+EW}$ is used for calculating numerator of $wt^{EW}$.

• 5. As in step (4), but $\rho _{\ell f}, {{\mathscr {K}}}_f, {{\mathscr {K}}}_l, {{\mathscr {K}}}_{\ell f}$ form-factors include box corrections. The ${{\mathscr {A}}}^{Born+EW}$ is used for calculating numerator of $wt^{EW}$.

Fig. 4

Top-left: line-shape distribution as generated with Powheg+MiNLO (blue triangles) and after reweighting introducing all EW corrections (red triangles). The two choices are barely distinguishable. Ratios of the line-shapes with gradually introduced EW corrections are shown in consecutive plots, where as a reference (black dashed line) respectively: (i) EW LO $\alpha (0)$ scheme (top-right), (ii) effective Born (LEP) (bottom-left) and, (iii) effective Born (LEP with improved norm.) (bottom-right), was used

After step (1) the sample is EW LO and QCD NLO, but with different EW scheme than used originally for events generation. Then steps (2)–(5) introduce EW corrections. Step (3) effectively changes $\alpha$ back to be close to $\alpha (M_Z^2)$, while steps (4)–(5) effectively shift back $v_f, v_l$ close to the values used in generation. Parameters for EW LO scheme used for event generation are already close to measured at the Z-pole. That is why we expect the total EW corrections to the generated sample to be roughly at the percent level only.

In the following, we will estimate how precise it would be to use effective Born approximation with LEP or LEP with improved norm. parametrisations instead of complete EW corrections. To obtain those predictions, re-weighting similar to step (1) listed above is needed, but in the numerator of $wt^{EW}$ the ${\mathscr {A}}^{Born}$ parametrisations as specified in the right two columns of Table 2 are used. For LEP with improved norm. the $\rho _{\ell ,f} = 1.005$ has to be included as well.

The important flexibility of the proposed approach is that $wt^{EW}$ can be calculated using $d \sigma _{Born}$ in different frames: $\cos \theta ^*$, Mustraal or Collins-Soper. For some observables, frame choice used for $wt^{EW}$ calculation is not numerically relevant at all; the simplest $\cos \theta ^*$ frame can be used. We show later an example, where only the Mustraal frame for the $wt^{EW}$ calculation leads to correct results.

8.1 The Z-boson line-shape

In the EW LO, the Z-boson line-shape, assuming that the constraint (15) holds, depends predominantly on $M_Z$ and $\Gamma _z$. The effects on the line-shape from EW loop corrections are due to corrections to the propagators: vacuum polarization corrections (running $\alpha$) and $\rho$ form-factor, which change relative contributions of the Z to $\gamma ^*$ and, the Z-boson vector to axial coupling ratio ($\sin ^2\theta _{eff}$). The above affects not only shape but also normalization of the cross-section. In the formulae (27) we do not use running Z-boson width, which remains fixed.

In Fig. 4 (top-left) distributions of generated and EW corrected line-shapes are shown. With the logarithmic scale, a difference is barely visible. With the following plots of the same Figure we study details. The ratios of the line-shape distributions with gradually introduced EW corrections are shown. For the reference distributions (ratio-histograms denominators) for the following three plots: (i) EW LO $\alpha (0)$ scheme, (ii) effective Born (LEP) and (iii) effective Born (LEP with improved norm.) are used. At the Z-pole, complete EW corrections contribute about 0.1% with respect to the one of effective Born (LEP with improved norm.). A use of events generated with EW LO matrix element but of different parametrisations significantly reduce the numerical size of missing EW corrections.

Table 3 details numerically EW corrections to the normalization (ratio of the cross-sections) integrated in the range $80< m_{ee} <100$ GeV and $89< m_{ee} < 93$ GeV. Results from EW weight with the $\cos \theta ^*$ definition of the scattering angle are shown. The total EW correction factor is about 0.965 for cross-section normalization and EW LO $\alpha (0)$ , while the total correction for the effective Born (LEP with improved norm.) is of about 1.001. In Table 4 results with $wt^{EW}$ calculated with different frames are compared. If Mustraal or Collins-Soper frames are used instead of $\cos \theta ^*$ for weight calculations, the differences are at most at the 5-th significant digit.

Table 3 EW corrections for cross-sections integrated over the specified mass windows. The EW weight is calculated with $\cos \theta ^*$

Table 4 EW corrections for cross-sections integrated over the mass window around Z-pole; $89< m_{ee} <$ 93 GeV. The EW weight is calculated with $\cos \theta ^*$, $\cos \theta ^{Mustraal}$ or $\cos \theta ^{CS}$

8.2 The $A_{FB}$ distribution

The forward-backward asymmetry for pp collisions reads

$$\begin{aligned} A_{FB} = \frac{\sigma (\cos \theta> 0) - \sigma (\cos \theta< 0)}{\sigma (\cos \theta > 0) + \sigma (\cos \theta < 0)}, \end{aligned}$$

(28)

where $\cos \theta$ of the Collins-Soper frame is used.

The EW corrections change $A_{FB}$, particularly around the Z-pole. In Fig. 5 (top-left), the $A_{FB}$ as generated (EW LO) and EW corrected is shown as a function of $m_{ee}$. In the following plots of this Figure, we study details. The $\Delta A_{FB} = A_{FB} - A_{FB}^{ref}$, with gradually introduced EW corrections to $A_{FB}$ is shown and compared with the following reference choices for $A_{FB}^{ref}$: (i) EW LO $\alpha (0)$ scheme, (ii) effective Born (LEP) and (iii) effective Born (LEP with improved norm.).

Complete EW corrections to predictions of EW LO $\alpha (0)$ scheme for $A_{FB}$ integrated around Z-pole give $\Delta A_{FB}$ = -0.03534. The EW correction $\Delta A_{FB}$ to predicition of effective Born (LEP with improved norm.), is only -0.00005. We observe that effective Born (LEP improved norm.) reproduces EW loop corrections precision better and $\Delta A_{FB}$ = -0.0001 in the full presented mass range. The remaining box corrections contribute around $m_{ee}=150$ GeV about -0.002 to $\Delta A_{FB}$.

Table 5 details numerically EW corrections, for $A_{FB}$ integrated over the $80< m_{ee} < 100$ GeV and $89< m_{ee} < 93$ GeV ranges. For calculating EW weight, the $\cos \theta ^*$ definition of the scattering angle was used. In Table 6 results obtained with $wt^{EW}$ calculated in different frames are compared. When the Mustraal or Collins-Soper frame is used instead of $\cos \theta ^*$, the differences are at most at the 5-th significant digit, similar as for the line-shape.

Fig. 5

Top-left: the $A_{FB}$ as generated with Powheg+MiNLO (blue triangles) and after reweighting introducing all EW corrections (red triangles). The two choices are barely distinguishable. The differences $\Delta A_{FB} = A_{FB} - A_{FB}^{ref}$, due to gradually introduced EW corrections are shown in consecutive plots, where as a reference (black dashed line) respectively: (i) EW LO $\alpha (0)$ scheme (top-right), (ii) effective Born (LEP) (bottom-left) and, (iii) effective Born (LEP with improved norm.) (bottom-right), was used

Table 5 The difference $\Delta A_{FB}$ in forward-backward asymmetry calculated in the specified mass window. The $\cos \theta ^{CS}$ is used to define forward and backward hemispheres. The EW weight is calculated from $\theta ^*$ definition of the scattering angle

Table 6 The difference $\Delta A_{FB}$ in forward–backward asymmetry around Z-pole, $m_{ee}$ = 89–93 GeV. The $\cos \theta ^{CS}$ is used to define forward and backward hemispheres. The EW weight is calculated respectively from $\cos \theta ^*$, $\cos \theta ^{Mustraal}$ or $\cos ^{CS}$

8.3 Effective weak mixing angles

The forward-backward asymmetry $A_{FB}$ at the Z-pole can be used as an observable for effective weak mixing Weinberg angles, dependent on the invariant mass of lepton pairs. We extend standard LEP definition of effective weak mixing angles to

$$\begin{aligned} \sin ^2\theta ^f_{eff}(s,t) = Re({\mathscr {K}}^f(s,t)) s^2_W + I^2_f(s,t), \end{aligned}$$

(29)

which is more suitable for LHC and for the off Z-pole regions. The flavour dependent effective weak mixing angles, calculated using: Eq. (29), EW form-factors of Dizet library, and $s^2_W=0.22352$ are shown on Fig. 6 as a function of the invariant mass of outgoing lepton pair and for $\cos \theta = 0.5$. The imaginary part of $I^2_f(s,t)$ is about $10^{-4}$ only. In Table 8 we display effective weak mixing angles averaged over specified mass windows.

The effective $\sin \theta _{eff}^f$ on the Z-pole, printed by Dizet is shown in Table 7. It is numerically slightly different than of Table 8, which is an average over mass window close to Z-pole. Note, that the observed very good agreement at the Z-pole between $A_{FB}$ predictions of effective Born with (LEP) or (LEP with improved norm.) parametrisations and fully EW corrected is not reflected for predictions of flavour dependent effective weak Weinberg angles. Effective Born (LEP) and (LEP with improved norm.) are parametrised with $s^2_W = 0.23152$, while Dizet library predicts leptonic effective weak mixing angle $\sin ^2\theta _{eff}^{\ell }(M_Z^2)$ = 0.23176 which is about $20 \cdot 10^{-5}$ different. Why then such a good agreement on $\Delta A_{FB}$ as seen on Fig. 5 bottom plots? Certainly this requires further attention.

Fig. 6

Effective weak mixing angles $\sin ^2\theta _{eff}^f(s,t)$ as a function of $m_{ee}$ and $\cos \theta$ = 0, without (left-hand plot) and with (right-hand plot) box corrections. The ${\mathscr {K}}^f(s,t)$ form-factor calculated using Dizet library and on-mass-shell $s^2_W=0.22352$ were used. Only the real part is shown, imaginary part of $I^2_f(s,t)$ is only about $10^{-4}$

Table 7 From the Dizet library printout: effective weak mixing angles and $\alpha (M_Z^2)$. For details of ZPAR parameter matrix definition see technical documentation of KKMC interface and DIZET library itself [2, 11]

Table 8 The effective weak mixing angles $\sin ^2 \theta _{eff}^{f}$, for different mass windows with/without box corrections. The form-factor corrections are averaged with realistic line-shape and $\cos \theta$ distribution

8.4 The $A_{4}$, $A_{3}$ angular coefficients

To complete the discussion on doubly-deconvoluted observables, we turn our attention to angular coefficients $A_{4}$ and $A_{3}$ (proportional to product of vector and axial couplings) and to EW corrections. The coefficients are calculated from the event sample with the moments methods [32] and in the Collins-Soper frame. The EW weight $wt^{EW}$ is used to introduce EW corrections and is calculated with the help of $\cos \theta ^*$, $\cos \theta ^{Mustraal}$ or $\cos \theta ^{CS}$ angles.

Similarly as for $A_{FB}$, the EW corrections change overall size and the shape of $A_4$ as a function of $m_{ee}$; particularly around the Z-pole. In Fig. 7 (top-right), the $A_4$ for generated sample (EW LO) and EW corrected is shown as a function of $m_{ee}$. In the following plots of the figure details are studied. The $\Delta A_4 = A_4 - A_4^{ref}$ with gradually introduced EW corrections is shown and compared with the following reference choices for $A_{4}^{ref}$: (i) EW LO $\alpha (0)$ scheme, (ii) effective Born (LEP) and (iii) effective Born (LEP with improved norm.). Conclusions are very similar as for previous $\Delta A_{FB}$ discussion. Note that $\Delta A_4$ and $\Delta A_{FB}$ scale approximately with the relation $A_4 = 8/3 A_{FB}$.

The analogous set of plots, Fig. 8, is prepared for $A_3$. In this case, only the Mustraal frame turned out to be adequate for $wt^{EW}$ calculation. Both the $\cos \theta ^*$ and $\cos \theta ^{CS}$ were unable to fully capture the effects of EW corrections.

The results for $\Delta A_{3}$ are collected in Table 9. The mass window $80< m_{ee} <100$ GeV and $p_T^{ee} < 30$ GeV are chosen. The estimation for $\Delta A_4$ differ little if $\cos \theta ^*$, $\cos \theta ^{CS}$ or $\cos \theta ^{Mustraal}$ is used for calculations of EW corrections. The $\Delta A_3$ is non-zero, as it should be, only if the $\cos \theta ^{Mustraal}$ is used in $wt^{EW}$ calculation. For $A_{4}$, multiplied by $\frac{8}{3}$ entries of Table 5 are good enough.

Fig. 7

Top-left: the $A_4$ as function of $m_{ee}$. Overlayed are generated and EW corrected $A_4$ predictions. These results are barely distinguishable. The differences $\Delta A_{4} = A_4 - A_4^{ref}$ due to gradually introduced EW corrections are shown in consecutive plots, where as a reference $A_4^{ref}$ (black dashed line) respectively (i) EW LO $\alpha (0)$ scheme (top-right), (ii) effective Born (LEP) (bottom-left) and (iii) effective Born (LEP with improved norm.) (bottom-right) was used

Fig. 8

Top-left: the $A_3$ as function of $m_{ee}$. Overlayed are generated and EW corrected $A_3$ predictions. These results are barely distinguishable. The differences $\Delta A_3 = A_3 - A_3^{ref}$ due to gradually introduced EW corrections are shown in consecutive plots, where as a reference $A_3^{ref}$ (black dashed line) respectively (i) EW LO $\alpha (0)$ scheme (top-right), (ii) effective Born (LEP) (bottom-left) and (iii) effective Born (LEP with improved norm.) (bottom-right) was used. In this case, the EW weight is calculated with $\cos \theta ^{Mustraal}$

9 Summary

In this paper we have shown how the EW corrections for double-deconvoluted observables at LHC can be evaluated using Improved Born Approximation. We have exploited a wealth of the LEP era results encapsulated in the Dizet library developed at that time. We have used that formalism to calculate and present numerically EW corrections for doubly-deconvoluted observables, such as Z-boson line-shape, forward-backward asymmetry $A_{FB}$, effective weak mixing angles or lepton direction angular coefficients.

We have followed largely discussions available in Dizet documentation. We have introduced the notion of the effective Born and explained how Monte Carlo events generated at NLO QCD can be transformed to reduced kinematics, of strong interaction lowest order, for the calculation of spin amplitudes $q {\bar{q}} \rightarrow Z/\gamma ^* \rightarrow \ell \ell$. This could be achieved thanks to properties of spin amplitudes discussed in [5, 6]. We explained how per-event weight $wt^{EW}$, can be build and used to attribute EW corrections to already generated events.

We have re-visited the notion of Effective Born with LEP (or with LEP of improved norm.) parametrisations where dominant parts of EW corrections are taken into accout with a redefinition of coupling constants. We have evaluated how well it works for observables of the paper. The discussed approach for treating EW corrections for Drell-Yan process in pp collisions has been implemented in the Tauola/TauSpinner package [9, 15] to be available starting from the forthcoming release.

Table 9 The $\Delta A_3$ shift of the $A_3$, due to EW corrections, averaged over $p_T^{ee} <$ 30 GeV and $80< m_{ee} < 100$ GeV ranges. The $\cos \theta ^{CS}$ is used for angular polynomials but for the EW weight calculation $\cos \theta ^*$, $\cos \theta ^{Mustraal}$ or $\cos \theta ^{CS}$ are used respectively

Once the formalism was explained, numerical results of EW corrections to the Z-boson line-shape, forward-backward asymmetries, lepton angular coefficients were presented. Results were obtained using Dizet for calculating EW form-factors and Tauola/TauSpinner for calculating respective EW weights of Improved Born Approximation or Effective Born with LEP (or with LEP improved norm.) parametrisations.

The choice of the version of EW library was dictated by the compatibility with the KKMC Monte Carlo [2], the program widely used at the LEP times. It relies on a published version of Dizet, thus suits the purposes of a reference point well. Also, omitted effects are rather small. In the future, the algorithm of TauSpinner can be useful to quantify the differences among distinct implementations of the electroweak sector.

The numerical studies with the updates to Dizet version 6.42 [12, 21] and with other, sometimes unpublished electroweak codes are left for the future work. One should stress the necessity of such future numerical discussion and updates, in particular due to the photonic vacuum polarization, e.g. as provided in Refs. [34, 35] but absent in the last published (or presently public) version of Dizet 6.42. This update is required already at LHC precision of Z-boson couplings measurements.

Fig. 9

The $A_4$ variation due to shifts induced with the presented in Appendix B options; as a function of $s^2_W$ (left-hand side) and as a function of $sin^2\theta _{eff}^l$ (right-hand side). The “FF $G_\mu$ varied”, FF $m_t$ varied” correspond to the case when form-factors were recalculated. Otherwise they were kept at nominal values

In many applications focused on challenges of strong interactions, electroweak corrections are receiving rather minimal attention and in particular Z boson fixed value width, or running only in proportion to the energy transfer, is used. This may be inappropriate for large s as found e.g. in [36]. TauSpinner can be used to evaluate numerical consequences of such approximation. Finally let us mention that presented implementation of EW corrections as per-even weight, was already found useful for experimental measurements [16] at LHC and for discussions during recent workshops, see e.g. Ref. [37].

Appendices

A Comment on technical details of TauSpinner EW effects implementation

Although the framework of Tauola/TauSpinner package [9, 15] has been used for numerical results presented in this paper, the code is not yet available with the public release but only in the private distribution and only partly in development version [38] which updates itself daily from our work repository. Tests and some of the code developments need to be completed. Once we achieve confidence the official stable version of the code will become public at [38] . Let us nonetheless list main points of the implementation which was already used to obtain numerical results:

• Pre-tabulated EW corrections: form-factors, vacuum polarization corrections in form of 2D root histograms or alternatively ASCII files of the KMMC project [2] were used to assure modularity and to enable graphic tests.

• Functions to calculate $\cos \theta ^*$, $\cos \theta ^{Mustraal}$, $\cos \theta ^{CS}$ from kinematics of outgoing final state (leptons and partons/jets) used for numerical results are already in part available in TAUOLA/TauSpinner/examples/ Dizet-example directory. The README file of that directory is gradually filled with technical details.

• Routine to initialize parameters of the Born function is provided. The SUBROUTINE INITWK of TAUOLA/

  src/tauolaFortranInterfaces/ tauola_extras.f has been copied and extended. It is available under the name INITWKSWDELT , with the following input:

  • $G_{\mu }$, $\alpha$, $M_Z$, s,

  • EW form-factors and vacuum polarization corrections,

  • $s^2_W$ and parameters for couplings variations $\delta _{s2W}$, $\delta _{V}$, see Section B for details.

• To calculate $d\sigma _{Born}$ and the $wt^{EW}$ the t_bornew function with flexible options for EW scheme and $\delta _{s2W}$, $\delta _{V}$, is prepared. It is used by TauSpinner library function default_nonSM_born(ID, S, cost, H1, H2, key) now.

• It is premature for complete documentation, but comments on the software used to obtain numerical results are in place.

Table 10 Dizet initialization parameters: masses and couplings

B How to vary $s^2_W$ beyond the EW LO schemes.

In the discussed EW scheme $(\alpha (0), G_\mu , M_Z)$, the $s^2_W$ is not directly available for fits. It is calculated from relation (16) of the Standard Model. One possibility to vary $s^2_W$, but stay within Standard Model framework is to vary some other constants which impact $s^2_W$. The candidates within Standard Model, which are also inputs to the Dizet library, are $G_{\mu }$ or $m_{t}$. From the simple estimates, to allow $\pm 100 \cdot 10^{-5}$ variation of $s^2_W$, those parameter will have to be varied far beyond their experimental ambiguities.¹⁵

Table 11 Dizet initialization flags. Unmodified comments taken from the KKMC code

Table 12 Dizet recalculated quantities available for the TauSpinner use. For details of the ZPAR table see Refs. [2, 11]

One can extend formulae for ${\mathscr {A}}^{Born+EW}$ (11) beyond the Standard Model too. Additional v-like contribution to Z-boson $v_{\ell }, v_f$ couplings can be introduced with $\delta _{S2W}$ or $\delta _{V}$ as presented later. Below few details and options on implementation into ${\mathscr {A}}^{Born+EW}$ amplitudes are given:

• optME = 1: introduce unspecified heavy particle coupling to the Z-boson, to modify fermions vector couplings

  $$\begin{aligned} v_{\ell } =&(2 \cdot T_3^{\ell } - 4 \cdot q_{\ell } \cdot (s^2_W + \delta _{S2W}) \cdot {{\mathscr {K}}}_{\ell }(s,t))/\Delta , \nonumber \\ v_f =&(2 \cdot T_3^f - 4 \cdot q_f \cdot (s^2_W + \delta _{S2W}) \cdot {{\mathscr {K}}}_f(s,t))/\Delta , \nonumber \\ vv_{\ell f} =&\frac{1}{v_{\ell } \cdot v_f} [ ( 2 \cdot T_3^{\ell }) (2 \cdot T_3^f) \nonumber \\&- 4 \cdot q_{\ell } \cdot (s^2_W+ \delta _{S2W}) \cdot {{\mathscr {K}}}_f(s,t)( 2 \cdot T_3^{\ell }) \nonumber \\&- 4 \cdot q_f \cdot (s^2_W+ \delta _{S2W}) \cdot {{\mathscr {K}}}_{\ell }(s,t) (2 \cdot T_3^f) \nonumber \\&+ (4 \cdot q_{\ell } \cdot s^2_W) (4 \cdot q_f \cdot s^2_W) {{\mathscr {K}}}_{\ell f}(s,t) \nonumber \\&+ 2 \cdot (4 \cdot q_{\ell }) (4 \cdot q_f) \cdot s^2_W \cdot \delta _{S2W}] \frac{1}{\Delta ^2} \end{aligned}$$

  (30)

  but do not alter

  $$\begin{aligned} \Delta = \sqrt{ 16 \cdot s^2_W \cdot (1 - s^2_W)} \end{aligned}$$

  (31)

  or any other ${\mathscr {A}}^{Born+EW}$ (11) couplings or calculations of the EW form-factors.

• optME = 2: recalculate $M_W$ for numerically modified $m_t$ or $G_\mu$ and modify accordingly Standard Model $s^2_W = 1 -M_W^2/M_Z^2$, for $s^2_W$ present in ${\mathscr {A}}^{Born+EW}$. The form-factors are (are not) recalculated.¹⁶ In total, 3 variants of this option were used for Fig. 9.

• optME = 3: similar as optME = 1 but redefine directly fermions vector couplings with $\delta _{V}$. We keep relative normalization (charge structure) of $\delta _{V}$ similar to $\delta _{S2W}$, to facilitate comparisons. Then

  $$\begin{aligned} v_{\ell } =&(2 \cdot T_3^{\ell } - 4 \cdot q_{\ell } \cdot (s^2_W \cdot {{\mathscr {K}}}_{\ell }(s,t) + \delta _{V} ))/\Delta , \nonumber \\ v_f =&(2 \cdot T_3^f - 4 \cdot q_f \cdot (s^2_W \cdot {{\mathscr {K}}}_f(s,t) + \delta _{V} ))/\Delta , \nonumber \\ vv_{\ell f} =&\frac{1}{v_{\ell } \cdot v_f} [ ( 2 \cdot T_3^{\ell }) (2 \cdot T_3^f) \nonumber \\&- 4 \cdot q_{\ell } \cdot (s^2_W \cdot {{\mathscr {K}}}_f(s,t) + \delta _{V} ) ( 2 \cdot T_3^{\ell }) \nonumber \\&- 4 \cdot q_f \cdot (s^2_W \cdot {{\mathscr {K}}}_{\ell }(s,t) + \delta _{V} ) (2 \cdot T_3^f) \nonumber \\&+ (4 \cdot q_{\ell } \cdot s^2_W) (4 \cdot q_f \cdot s^2_W) {{\mathscr {K}}}_{\ell f}(s,t) \nonumber \\&+ 2 \cdot (4 \cdot q_{\ell }) (4 \cdot q_f) \cdot s^2_W \cdot \delta _{V} ]\ \frac{1}{\Delta ^2}. \end{aligned}$$

  (32)

  The $\delta _{V}$ shift is almost equivalent to $\delta _{S2W}$ shift, but affects couplings in a (s, t) independent manner.

Even though discussion of $s^2_W$ variation necessary for fits, is generally out of scope of the present paper and it can not be now exhausted, let us provide some numerical results to illustrate stability of the method.¹⁷ The variations for $A_4(M_Z)$ are presented in Figure 9: as a function of $s^2_W$ on the left-hand side plot and as a function of $sin^2\theta _{eff}^l$ on the right-hand side plot. It is very reassuring, that all presented optME methods lead to the same slope of the $A_4(M_Z)$ as a function of $sin^2\theta _{eff}$. Very similar curve could be presented for $A_{FB}$, which would be scaled by $\frac{3}{8}$ with respect to $A_4$ only.

C Initialization of the Dizet library

There is a wealth of initialization constants and options available for Dizet library. The documentation of that program and of its interface for KKMC, explains options available for the TauSpinner users as well. Tables 10 and 11 recall available Dizet initialization, Table 12 lists calculated by Dizet quantities for the use in TauSpinner library.

In the present work, we have relied on the Dizet library version as installed in the KKMC Monte Carlo [8] and used at a time of LEP 1 in detector simulations. Already for the data analysis and in particular for final fits [1], further effects of minor, but non-negligible numerical impact were taken into account. Gradually, effects such as improved top contributions [39] or better photonic vacuum polarization [23], were taken into account. This has to be updated for Dizet library too.

Such update is of importance also for the KKMC project itself because of forthcoming applications for the Future Circular Collider or for LHC [40].